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Cyclic numbers are related to the recurring digital representations of unit fractions. A cyclic number of length L is the digital representation of 1/(L + 1). Conversely, if the digital period of 1/p (where p is prime) is p − 1, then the digits represent a cyclic number. For example: 1/7 = 0.142857 142857...
A cyclic number [1] [2] is a natural number n such that n and φ(n) are coprime. Here φ is Euler's totient function. An equivalent definition is that a number n is cyclic if and only if any group of order n is cyclic. [3] Any prime number is clearly cyclic. All cyclic numbers are square-free. [4] Let n = p 1 p 2 …
A number n is called a cyclic number if Z/nZ is the only group of order n, which is true exactly when gcd(n, φ(n)) = 1. [13] The sequence of cyclic numbers include all primes, but some are composite such as 15. However, all cyclic numbers are odd except 2. The cyclic numbers are:
Cyclic number, a number such that cyclic permutations of the digits are successive multiples of the number; Cyclic order, a ternary relation defining a way to arrange a set of objects in a circle; Cyclic permutation, a permutation with one nontrivial orbit; Cyclic polygon, a polygon which can be given a circumscribed circle; Cyclic shift, also ...
For powers of 2 the factor (/) is not cyclic unless k = 0, 1, 2, but factors into cyclic groups as described above. The order of the group φ ( n ) {\displaystyle \varphi (n)} is the product of the orders of the cyclic groups in the direct product.
In algebra, a cyclic division algebra is one of the basic examples of a division algebra over a field and plays a key role in the theory of central simple algebras. Definition [ edit ]
A cyclic group, C n is the group of rotations of a regular n-gon, that is, n elements equally spaced around a circle. This group has φ(d ) elements of order d for each divisor d of n, where φ(d ) is the Euler φ-function, giving the number of natural numbers less than d which are relatively prime to d.
In mathematics, a real number is a number that can be used to measure a continuous one-dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. [a] Every real number can be almost uniquely represented by an infinite decimal expansion. [b] [1]